BGG :: BGG

# BGG -- Bernstein-Gel'fand-Gel'fand correspondence

## Description

The Bernstein-Gel'fand-Gel'fand correspondence is an isomorphism between the derived category of bounded complexes of finitely generated modules over a polynomial ring and the derived category of bounded complexes of finitely generated module over an exterior algebra (or of certain Tate resolutions). This package implements routines for investigating the BGG correspondence.

More details can be found in Sheaf Algorithms Using Exterior Algebra.

## Version

This documentation describes version 1.4.2 of BGG.

## Source code

The source code from which this documentation is derived is in the file BGG.m2.

## Exports

• Functions and commands
• Methods
• "beilinson(Matrix,PolynomialRing)" -- see beilinson -- Vector bundle map associated to the Beilinson monad
• "bgg(ZZ,Module,PolynomialRing)" -- see bgg -- the ith differential of the complex R(M)
• "cohomologyTable(CoherentSheaf,ZZ,ZZ)" -- see cohomologyTable -- dimensions of cohomology groups
• "cohomologyTable(Matrix,PolynomialRing,ZZ,ZZ)" -- see cohomologyTable -- dimensions of cohomology groups
• directImageComplex(ChainComplex) -- direct image of a chain complex
• directImageComplex(Matrix) -- map of direct image complexes
• directImageComplex(Module) -- Complex representing the direct image
• "projectiveProduct(Matrix,List)" -- see projectiveProduct -- Makes a product of projective spaces and a system of parameters
• "projectiveProduct(Ring,List)" -- see projectiveProduct -- Makes a product of projective spaces and a system of parameters
• "pureResolution(Matrix,List)" -- see pureResolution -- creates a pure resolution as an iterated direct image
• "pureResolution(Ring,List)" -- see pureResolution -- creates a pure resolution as an iterated direct image
• "pureResolution(ZZ,List)" -- see pureResolution -- creates a pure resolution as an iterated direct image
• "pureResolution(ZZ,ZZ,List)" -- see pureResolution -- creates a pure resolution as an iterated direct image
• "symExt(Matrix,PolynomialRing)" -- see symExt -- the first differential of the complex R(M)
• "tateResolution(Matrix,PolynomialRing,ZZ,ZZ)" -- see tateResolution -- finite piece of the Tate resolution
• "universalExtension(List,List)" -- see universalExtension -- Universal extension of vector bundles on P^1
• Symbols
• Exterior -- dual exterior algebra cached in a polynomial ring
• Regularity -- Option for directImageComplex

## For the programmer

The object BGG is .